You need to change the way math classes think of proofs. Every times I have seen a proof in math class, they managed to take a beautiful and elegant argument, and distill it down to the least number of characters that formally proves the statement.
Presently, I am doing research work in (applied) chryptography, which as you might imagine involves reading a fair amount of modern math papers. All but one of the papers I read were easier to understand than the textbook proof for Pythagorean's thuerom.
I agree. More motivations, intuitions, insights behind fewer proofs would be more useful than lots of proofs, if the goal is to teach mathematical thinking, not to teach particular math results.
I think proofs are compressed because of lack of margins. Don't laugh! Authors seem to think they need to cover lots of essential results, and if you do motivations, intuitions, insights for all of them, already long math books will be impossible to hold in your hands. In my opinion the obvious solution is to omit most proofs and do far more non-formal discussion around important proofs. Which won't happen because their goal is to teach particular math results, not mathematical thinking, contrary to what they say.
